Note:
Brachistochrone is a term from Greek meaning "shortest time." The
special property of a brachistochrone is the fact that a bead sliding
down
a brachistochrone-shaped frictionless wire will take a shorter
time to reach the bottom than with a wire curved into any other
shape. In addition, since a brachistochrone is also a tautochrone,
the bead will take the same amount of time to reach the bottom
no matter
how
high
or low
the
release point.
Key Formula
x=r(θ−sinθ),y=r(1−cosθ)
Where:
r = Radius of the generating circle of the cycloid
θ = Parameter (angle) that traces out the curve, ranging from 0 to 2π for one arch
x = Horizontal position along the curve
y = Vertical position (measured downward from the starting point)
Worked Example
Problem: A bead slides from rest down a frictionless brachistochrone wire from point A at the origin to point B located at the bottom of one half-arch of a cycloid with generating circle radius r = 1 m. Find the time of descent. Take g = 9.8 m/s².
Step 1: The time of descent along a brachistochrone from the top of a cycloid arch to the lowest point is given by a known result from calculus of variations.
T=πgr
Step 2: Substitute r = 1 m and g = 9.8 m/s² into the formula.
T=π9.81=π0.10204
Step 3: Evaluate the square root and multiply by π.
T=π×0.3194≈1.003 s
Answer: The bead reaches the bottom in approximately 1.00 seconds.
Frequently Asked Questions
Why isn't a straight line the fastest path between two points?
A straight line is the shortest distance, but not the shortest time under gravity. The brachistochrone dips steeply at first, so the bead accelerates quickly and reaches a high speed early on. Even though the total path length is longer than a straight line, the higher average speed more than compensates, resulting in a shorter travel time.
What is the difference between a brachistochrone and a tautochrone?
Both are inverted cycloids, but they describe different properties. The brachistochrone is the curve of fastest descent between two given points. The tautochrone is the curve on which the time to slide to the bottom is the same regardless of the starting point. Remarkably, the same cycloid shape satisfies both properties.
Brachistochrone vs. Tautochrone
Both are inverted cycloids. The brachistochrone minimizes the travel time from one specific point to another — it answers 'what shape is fastest?' The tautochrone guarantees equal travel time from any starting height to the bottom — it answers 'what shape gives the same time regardless of release point?' The remarkable coincidence is that the same curve, the cycloid, solves both problems.
Why It Matters
The brachistochrone problem, posed by Johann Bernoulli in 1696, was one of the earliest problems in the calculus of variations — a branch of mathematics that optimizes entire curves or functions rather than single values. Its solution helped develop techniques now used in physics (Lagrangian mechanics, general relativity), engineering (optimal control, trajectory design), and economics. The problem also illustrates a counterintuitive principle: the shortest path and the fastest path are generally not the same.
Common Mistakes
Mistake: Assuming the fastest path between two points must be a straight line.
Correction: A straight line minimizes distance, not time. Under gravity, the brachistochrone (an inverted cycloid) is faster because it allows the object to gain speed quickly by dropping steeply at the start.
Mistake: Confusing the brachistochrone with an ordinary parabolic trajectory.
Correction: A parabola describes the path of a freely thrown projectile. The brachistochrone is a cycloid, which is a fundamentally different curve arising from constrained motion along a wire, not free flight.
Related Terms
Cycloid — The specific curve shape of the brachistochrone
Tautochrone — Same cycloid curve with equal-time property
